All equilibrium methods considered in this section involve the distribution of substances between two phases that are insoluble in one another. As an example, consider the two immiscible liquids benzene and water. If a colored compound is placed in the water and the two phases are mixed, color appears in the benzene phase, and the intensity of the color in the water phase decreases. These color changes continue to occur for a certain time, beyond which no macroscopic changes take place, no matter how long or vigorously the two phases are mixed. Because the dye is soluble in the benzene as well as in the water, the dye is extracted into the benzene at the start of the mixing. But, just as the dye tends to move into the benzene phase, so it also tends to be dissolved in the aqueous phase. Thus dye molecules move back and forth across the liquid-liquid interface. Eventually, a condition is reached such that the tendencies of the dye to pass from benzene to water and from water to benzene are equal, and the concentration of the dye (as measured by the intensity of its color) is constant in the two phases. This is the condition of equilibrium. Note that this is static from a macroscopic point of view. On a molecular level it is a dynamic process, however, for many molecules continue to pass through the liquid-liquid interface (although of equal number in both directions).

The condition of equilibrium in this example can be described in terms of the distribution coefficient, K, by the equationin which the concentrations in the equilibrium state are considered. For K = 1, there are equal concentrations of the dye in the two phases; for K > 1, more dye would be found in the benzene phase at equilibrium. At K = 100, 99.01 percent is in the benzene, and only 0.99 percent is in the water (assuming equal volumes of the two liquids). For certain purposes, this condition might be considered to represent essentially complete removal of the dye from the water, but more often K = 1,000 is selected (i.e., 99.9 percent removal). Depending on the phases and conditions, it is often possible to achieve a K value of 1,000 or more.

Separation results when the distribution coefficient values for two substances (e.g., two dyes) differ from one another. Consider a case in which K = 100 for one substance and K = 0.01 for a second substance: then, upon reaching equilibrium, 99 percent of the former substance will be found in the benzene phase, and 99 percent of the latter substance will be found in the aqueous phase. It is clear that this sample is rather easily separated by liquid-liquid distribution. The ease of the separation thus depends on the ratio of the two distribution coefficients, α (sometimes called the separation factor):in which K₁ and K₂ are the respective distribution coefficients of components 1 and 2. In the above example, α = 10,000. In many other cases, α can be extremely small, close to unity (α is defined such that it is always unity or greater): then separation is difficult, requiring very efficient methods. Part of the art of separations is finding conditions that produce large separation factors of pairs of substances.

In Table 1 most of the important chemical equilibrium separation methods are subdivided in terms of the two insoluble phases (gas, liquid, or solid). A supercritical fluid is a phase that occurs for a gas at a specific temperature and pressure such that the gas will no longer condense to a liquid regardless of how high the pressure is raised. It is a state intermediate between a gas and a liquid. The example previously cited involved extraction (liquid-liquid). The other methods are described below.